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The Emergence and Interpretation of Probability in Bohmian Mechanics1

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The Emergence and Interpretation of Probability in Bohmian Mechanics1 The Emergence and Interpretation of Probability in Bohmian Mechanics1 The Bohm interpretation of quantum mechanics is capable of illustrating, by itself, virtually every philosophical and foundational conundrum associated with physical probability. One reason this is true is that the Bohm interpretation comes in many forms, both stochastic and deterministic. The other reason is that quantum mechanics is to Bohmian mechanics roughly as statistical mechanics is to classical mechanics; hence the notorious problems in the foundations of statistical mechanics are reprised within the foundations of Bohmian mechanics. The present paper is an opinionated survey of this literature. In it I focus on the meaning of Born’s rule in a Bohmian universe. After considering various rationales for this rule, I settle on one based on the law of large numbers as the best bet. This option leaves open how the probabilities in this result ought to be interpreted, or so I shall argue. When delivering an interpretation of these probabilities, the history of probability warns us of a number of pitfalls. I show how these pitfalls manifest themselves in the Bohmian case and then show that at least one class of interpretations seems to successfully navigate these problems. 1. Bohmian Mechanics and the Distribution Postulate Bohmian mechanics is an empirically adequate and logically consistent quantum theory of non-relativistic phenomena. To my mind it provides the most natural answer to the notorious measurement problem. The measurement problem tells us that the wave function evolving according to the Schrödinger equation either can’t be correct or it can’t be the whole story. Collapse theories answer the problem by denying the evolution is always via the Schrödinger equation. Bohmian theories answer by denying the completeness of the wavefunction description of the world—there are also particles, or “beables” in Bell’s terminology. Originally devised by de Broglie in 1927, it was subsequently rediscovered and improved by Bohm in 1952. More recently, many researchers have developed the approach further, especially the group led by Dürr, Goldstein and Zanghi (DGZ).2 For the purposes of this article, we shall limit ourselves to a brief sketch of the non-relativistic theory. The reader is encouraged to turn to the above references for treatments of spin, identical particles, quantum field theory, and more. Bohmian mechanics is a theory of N point particles evolving in R3. Each particle (indexed by i) obeys a first-order equation of motion: 1 Many thanks for comments and discussions are due to Jeff Barrett, Detlef Dürr, Sheldon Goldstein, Roman Frigg, Carl Hoefer, Barry Loewer and Tim Maudlin. 2 For a selection of their work, the reader can visit www.bohmian-mechanics.net for papers and references. Tumulka 2004 is a useful introduction to Bohm’s theory, and DGZ 1992 is a rigorous development of the theory. dx " #(x (t)...x (t)) i = h Im i 1 N (1) dt m #(x (t)...x (t)) i 1 N where ‘Im’ means imaginary part, m is the mass of the particle, and Ψ is a time- dependent, complex-valued function on configuration space R3N. The wave function Ψ ! evolves according to the familiar Schrödinger equation: N 2 "# h ih = $& % i •% i# + V (q1...qN )# (2) "t i=1 2mi 3N where V is the potential energy. Here qi are the variables in R upon which Ψ depends. The system given by (1) and (2) describes a deterministic theory of the universe, one for ! which global existence and uniqueness can be proved for almost all initial conditions. Relevant to us later, the theory is also time-reversal invariant and (for most states) Poincaré recurrent. That is, for every history there is also the time-reversed history given by the transformations t→-t and Ψ→Ψ*; and for every initial condition (x1(0)..xN(0),Ψ(0)), the time development by (1) and (2) will eventually take the system arbitrarily close to this state again. So far, nothing has been said of probabilities, yet the empirical heart of quantum mechanics is the fact that subsystems of the universe (for instance, actual ensembles measured in labs) are probabilistically distributed according to Born’s rule. The probability density for finding particles in configuration q≡(q1…qN) at time t, i.e., ρ(q,t), is equal to |Ψ(q,t)|2. To be as successful as ordinary quantum mechanics, Bohmian mechanics must provide a reason why betting in accord with Born’s rule is a rational strategy. That is, the Bohmian must explain why particles upon measurement actually will be distributed according to |Ψ(q,t)|2. The Bohmian needs to explain the origin of Born’s rule. Care needs to be taken in understanding Born’s rule in a Bohmian world. The wavefunction Ψ in (2) is the wavefunction of the universe, whereas the wavefunction empirically tested by Born’s rule is a function of a subsystem of the universe. This distinction is important for what follows. Fortunately, the Bohmian can speak of a subsystem or effective wavefunction ψ when—roughly—enough “decoherence” has occurred between the system and its environment (see DGZ 1992).3 The Bohmian will 3 Let Qt be the actual configuration of particles in the universe at a time. In a composite system Qt=(Xt,Yt), where X is the actual subsystem of interest and Y is the actual environment. Then a natural definition for a subsystem’s wavefunction is what DGZ call a conditional wavefunction ψt(x)= Ψt(x,Yt), where we calculate the universal wavefunction in the actual configuration of the environment. The conditional wavefunction will not in general evolve according to the Schrödinger equation. However, when certain further conditions are met (corresponding to the universal wavefunction evolving into a wide separation of components in the configuration space thus understand Born’s rule in terms of ψ, not Ψ. Since position measurements reveal the positions of actual particles for the Bohmian, the success of Born’s rule implies that the Bohmian particle positions of measured subsystems of the universe be randomly distributed according to |ψ(q,t)|2. Thus, the Bohmian must assume, derive or otherwise make plausible the claim that the position probability density is given by ρ(q,t) = |ψ(q,t)|2 (3) whenever a measurement is made. Let’s call (3) the distribution postulate. With (3) the Bohmian can explain the occurrence of frequencies in accord with Born’s rule and all that follows from that, e.g., why quantum mechanics works, why the uncertainty principle holds, why the ‘no superluminal signaling’ theorem obtains, and more. Without (3) or something close to (3), the Bohmian runs a serious explanatory deficit. The project of justifying (3) is often packaged as being very simple due to the following mathematical fact. The Schrödinger equation implies a continuity equation for |Ψ|2. If we substitute ρ(q,t) = |Ψ (q,t)|2 into the continuity equation then we get "# "(#x˙ ) + = 0 (4) "t "x Equation (4) will take any initial distribution of particles such that ρ(q,0) = |Ψ(q,0)|2 holds and evolve it in time t to the distribution ρ(q,t) = |Ψ(q,t)|2. The dynamics, in other ! 2 words, preserves the probability distribution ρ = |Ψ| . The project of justifying (3) is then advertised as simply the problem of finding some reason to think ρ(q,t) = |Ψ(q,t)|2 at some time or other, for one is then guaranteed that it will always hold. However, despite being repeated many times, this way of putting matters is misleading.4 What we want explained is why a system of particles should have a probability density given by the effective wavefunction, not the universal wavefunction. The effective wavefunction is clearly the relevant object in Bohmian mechanics, for it is the object corresponding to the wavefunction in the usual use of Born’s rule within orthodox quantum measurement theory. Yet the effective wavefunction does not always obtain. It evolves according to the Schrödinger equation when it exists, but the conditions for its existence do not always obtain. So we cannot make use of the above guarantee. That the dynamics preserves ρ = |Ψ|2 does not imply that the dynamics preserves ρ(q,t) = |ψ(q,t)|2. of the entire system--see DGZ for details) it will evolve according to the Schrödinger equation. When this is the case it is called an effective wavefunction. Note that an effective wavefunction obtains whenever the orthodox formalism assigns a wavefunction. 4 Thanks go to Tim Maudlin and Sheldon Goldstein for this point. The primary reason for this mistake, I suspect, is that prior to DGZ 1992 Bohm’s theory was not rigorously presented as a universal theory. In hindsight, Bohmians were unwittingly talking about universal wavefunctions when they should have been talking about effective wavefunctions. The preservation of ρ = |Ψ|2 is relevant and perhaps is a necessary condition for (3) obtaining, but it is not sufficient. 2. The Analogy with Classical Statistical Mechanics Einstein famously wanted the statistical aspects of quantum theory to take “an approximately analogous position to the statistical mechanics within the framework of classical mechanics” (1949, p. 672). Since Bohmian mechanics is a deterministic and more fundamental theory than quantum mechanics, it is in the perfect position to realize this goal. Let’s see how, first by comparing Bohmian mechanics with classical mechanics, and then by comparing quantum mechanics (as seen through Bohmian eyes) with classical statistical mechanics. Classical mechanics is also a theory of N point particles evolving on R3. In contrast to Bohmian mechanics, wherein one requires an initial wavefunction and initial locations, here one requires an initial position and initial momentum.
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